We explain the construction of Forcer, a Form program for the reduction of four-loop massless propagator-type integrals to master integrals. The resulting program performs parametric IBP reductions ...similar to the three-loop Mincer program. We show how one can solve many systems of IBP identities parametrically in a computer-assisted manner. Next, we discuss the structure of the Forcer program, which involves recognizing reduction actions for each topology, applying symmetries, and transitioning between topologies after edges have been removed. This part is entirely precomputed and automatically generated. We give examples of recent applications of Forcer, and study the performance of the program. Finally we demonstrate how to use the Forcer package and sketch how to prepare physical diagrams for evaluation by Forcer.
Program title: Forcer.
Program files doi:http://dx.doi.org/10.17632/d4jy7mfjxf.1
Licensing provisions: GNU General Public License, version 3.
Programming language: FORM. The generating scripts are written in Python.
Nature of problem: Some physical quantities in perturbative quantum field theories require the evaluation of dimensionally regularized massless propagator-type Feynman integrals. This can be achieved by reducing the integrals in the problem to a set of master integrals and then substituting their values. At the four-loop level, such a reduction becomes highly non-trivial and very time-consuming. One needs an efficient program for the reduction.
Solution method: The program recursively applies pre-derived parametric reduction rules for four-loop massless propagator-type Feynman integrals. Many rules were derived from topology substructures and were automatically generated by Python code. Some special cases were carefully optimized in a computer-assisted manner.
Restrictions: Limitations come from the fact that reducing too complicated integrals requires too much CPU time and storage space. If rational coefficients become too large to be handled in FORM, then one needs to consider using truncated expansions.
A growing body of evidence suggests that seismicity is seasonally modulated in a variety of tectonic environments. Identifying cyclic variations in seismicity improves our understanding of the ...physics of earthquake triggering. We explored seasonal modulation of crustal seismicity in San‐in district, southwest Japan, using a new method that adopts uncertainties derived from a probability‐based declustering procedure. We determined that semiannual variation in background seismicity rate, which increases in spring and autumn, is statistically significant from 1980 to 2017. The frequency distribution of large historic and modern earthquakes shows a similar pattern to recent background seismicity, suggesting that seismicity in San‐in district has shown seasonal variations for over 150 years. These observations can be explained by increasing pore pressure within fault zones, caused by precipitation in autumn and decreasing surface mass due to snow melting in spring.
Plain Language Summary
Seismic activity is seasonally modulated by surface processes. To explore earthquake seasonality, it is necessary to choose only independent earthquakes from entire seismicity catalog; this selection process is called earthquake declustering. Here, we introduce a new statistical declustering method to evaluate the significance of seasonal modulation of shallow seismicity, which includes an estimate of the uncertainty associated with earthquake declustering. We apply this technique to shallow crustal seismicity in San‐in district, southwest Japan. The rate of independent earthquakes from 1980 to 2017 shows significant semiannual variation, with increases in spring and autumn. The distribution of large historic and modern earthquakes shows similar variations, suggesting that seismicity in San‐in district has been related to the seasons for at least 150 years. This seasonality can be attributed to heavy precipitation in autumn and snow melting in spring. Seasonal modulation of small earthquakes should be monitored to investigate temporal changes in the probabilities of large earthquakes in San‐in district.
Key Points
A new method evaluates the significance of earthquake seasonality adopting uncertainty derived from a stochastic declustering scheme
The background seismicity rate of San‐in district in Japan shows significant semiannual variation that increases in spring and autumn
Seasonal variations in seismicity can be explained by heavy precipitation in autumn and snow melting in spring
We have computed the contributions of the quartic Casimir invariants to the four-loop anomalous dimensions of twist-2 spin-N operators at N≤16. The results provide new information on the structure of ...the next-to-next-to-next-to-leading order (N3LO) splitting functions Pik(3)(x) for the evolution of parton distributions, and facilitate approximate expressions which include the quartic-Casimir contributions to the (light-like) gluon cusp anomalous dimension. These quantities turn out to be closely related, by a generalization of the lower-order ‘Casimir scaling’, to the corresponding quark results. Using these findings, we present an approximate result for the four-loop gluon cusp anomalous dimension in QCD which is sufficient for phenomenological applications.
A
bstract
We have computed the five-loop corrections to the scale dependence of the renormalized coupling constant for Quantum Chromodynamics (QCD), its generalization to non-Abelian gauge theories ...with a simple compact Lie group, and for Quantum Electrodynamics (QED). Our analytical result, obtained using the background field method, infrared rearrangement via a new diagram-by-diagram implementation of the R
*
operation and the Forcer program for massless four-loop propagators, confirms the QCD and QED results obtained by only one group before. The numerical size of the five-loop corrections is briefly discussed in the standard
M
S
¯
scheme for QCD with
n
f
flavours and for pure SU(
N
) Yang-Mills theory. Their effect in QCD is much smaller than the four-loop contributions, even at rather low scales.
We have computed the fourth-order nf2 contributions to all three non-singlet quark–quark splitting functions and their four nf3 flavour-singlet counterparts for the evolution of the parton ...distributions of hadrons in perturbative QCD with nf effectively massless quark flavours. The analytic form of these functions is presented in both Mellin N-space and momentum-fraction x-space; the large-x and small-x limits are discussed. Our results agree with all available predictions derived from lower-order information. The large-x limit of the quark–quark cases provides the complete nf2 part of the four-loop cusp anomalous dimension which agrees with two recent partial computations.
We have computed the four lowest even-N moments of all four splitting functions for the evolution of flavour-singlet parton densities of hadrons at the fourth order in the strong coupling constant ...αs. The perturbative expansion of these moments, and hence of the splitting functions for momentum fractions x≳0.1, is found to be well behaved with relative αs-coefficients of order one and sub-percent effects on the scale derivatives of the quark and gluon distributions at αs≲0.2. More intricate computations, including other approaches such as the operator-product expansion, are required to cover the full x-range relevant to LHC analyses. Our results are presented analytically for a general gauge group for detailed checks and validations of such future calculations.
A
bstract
We have computed the self-energies and a set of three-particle vertex functions for massless QCD at the four-loop level in the
M
S
¯
renormalization scheme. The vertex functions are ...evaluated at points where one of the momenta vanishes. Analytical results are obtained for a generic gauge group and with the full gauge dependence, which was made possible by extensive use of the Forcer program for massless four-loop propagator integrals. The bare results in dimensional regularization are provided in terms of master integrals and rational coefficients; the latter are exact in any space-time dimension. Our results can be used for further precision investigations of the perturbative behaviour of the theory in schemes other than
M
S
¯
. As an example, we derive the five-loop beta function in a relatively common alternative, the minimal momentum subtraction (MiniMOM) scheme.
We have extended our previous computations of the even-N moments of the flavour-singlet four-loop splitting functions to N=12 for the pure-singlet quark case and N=10 for all other cases. These ...results, obtained using physical quantities in inclusive deep-inelastic scattering, have been and will be used to validate conceptionally much more challenging determinations of these splitting functions from off-shell operator matrix elements (OMEs). For the quark-gluon and gluon-gluon splitting functions, which have yet to be computed to higher N using OMEs, we construct approximations based on our moments and endpoint constraints, where we present new large-x results for the gluon-gluon case. These approximations facilitate an approximate N3LO evolution of parton distributions which are sufficiently accurate outside the region of small momentum fractions x.
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